lewisa asked "What's the difference between the mandarin language and the chinese one? Is it a dialect or simply a complete language with it's vocabulary and a special grammar?"

A short answer is: Mandarin Chinese is a subset of Chinese which forms the basis of the standard and official language (both speech and written forms). Mandarin is sometimes called a dialect of Chinese, but it is more accurate to reserve the term "dialect" for local language varieties (and the Mandarin speaking regions of China include many) and refer to Mandarin as a regional variety of Chinese (avoiding the controversial issue of whether the other regional varieties of Chinese are separate languages or not; linguists generally treat them as separate languages but with some caveats because of the shared writing system). Mandarin has its own vocabulary and special grammar different from the other regional varieties of Chinese.

A longer answer:

In English, the term "Chinese" can be used broadly (referring to all the varieties of speech and written language that share the same heritage of classical Chinese, and which are now quite divergent) or narrowly (to refer to "standard Chinese," in other words the dominant and official variety of speech, as well as the style of writing and reading that is based on it). The narrow reference of Chinese is also called Mandarin, when the distinction is important. The broad meaning covers the native language varieties of about 95% of the population in the People's Republic of China, while the narrow meaning refers to the variety that is natively spoken by about 70% of the population, although most of the other 30% speak Mandarin as a second language (they call it Pǔtōnghuà 普通话) and mostly use Mandarin as the basis of their writing. The 30% of mainland Chinese who are not native speakers of Mandarin Chinese includes somewhat less that 5% of the population who are not Han Chinese at all, the so-called national minorities, and whose language heritage usually includes a non-Chinese language which they may speak natively (the grandparents generation spoke non-Chinese languages, although there may be ongoing language shift among children in the 21st century). It also includes 25% of mainland China's population who are Han Chinese but whose heritage includes a regional variety of Chinese (called fāngyán 方言, which is sometimes translated as "dialect" but the term is ambiguous when applied to China). The non-Mandarin regional varieties of Chinese are traditionally divided into 8 groupings, which include the well-known Wu, Cantonese, Min (which includes Hokkien) and Hakka, although recent research is proposing to split off some varieties, I don't think there is a new consensus yet on how many. So you use the term Mandarin Chinese or just Mandarin when you want to refer to the native language of the 70%, which has status as an official language in all of mainland China, as well as in Taiwan, Hong Kong SAR and Singapore. If you want to use the term Chinese broadly (the 95%), you refer to all the historically-related language varieties (fāngyán) of the Han Chinese. Most Mandarin native-speakers cannot understand the native speech of the other 7 (or more) regional varieties, but they all share the same writing system (both the classical writing tradition based on obsolete speech varieties which, more or less, are the ancestors of all the regional varieties of Chinese, as well as the contemporary writing style (called vernacular Chinese or báihuà 白花) which is based on Mandarin Chinese. Most Chinese who are not native speakers of Mandarin can understand Mandarin (but not the other six or more regional varieties of Chinese) because they learn it in school and through the mass media.

- Fred, a linguistics student trying to learn written Chinese by independent study